Grammarly has a terrible privacy policy, so you are right to be cautious. Unfortunately I don't have any good alternatives to offer as I only use spellcheck myself.
This one is recommended by Mozilla
https://addons.mozilla.org/en-CA/firefox/addon/languagetool/
It DOES still need to send data somewhere to check
Your privacy is important to us: By default, this extension will check your text by sending it to languagetool.org over a securely encrypted connection. No account is needed to use this extension. We don't store your IP address. See https://languagetool.org/privacy/ for our privacy policy.
I don't want to confirm details I don't know, so someone else should probably explain more on if this is good/bad
It's on my to-do list, but you can also spin up your own language tool instance so that your data never leaves your house, since it is open source: https://github.com/languagetool-org/languagetool
If you have a homeserver it can go there, otherwise you can also run it on your computer although I am not sure how much RAM it will use.
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You can run LanguageTool locally. While it isn't as great as the paid version, I use this to check nearly everything I write for work in my native language, and in the other languages I speak
https://caderek.github.io/gramma/ is a cli spellchecker that has the option of installing a LT server locally. Not ideal if you are writing things with Pages/Word/etc., but a possible backup.
You can run your own instance of this?
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Very true.